// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_EIGENSOLVER_H
#define EIGEN_EIGENSOLVER_H

#include "./RealSchur.h"

namespace Eigen {

/** \eigenvalues_module \ingroup Eigenvalues_Module
  *
  *
  * \class EigenSolver
  *
  * \brief Computes eigenvalues and eigenvectors of general matrices
  *
  * \tparam _MatrixType the type of the matrix of which we are computing the
  * eigendecomposition; this is expected to be an instantiation of the Matrix
  * class template. Currently, only real matrices are supported.
  *
  * The eigenvalues and eigenvectors of a matrix \f$ A \f$ are scalars
  * \f$ \lambda \f$ and vectors \f$ v \f$ such that \f$ Av = \lambda v \f$.  If
  * \f$ D \f$ is a diagonal matrix with the eigenvalues on the diagonal, and
  * \f$ V \f$ is a matrix with the eigenvectors as its columns, then \f$ A V =
  * V D \f$. The matrix \f$ V \f$ is almost always invertible, in which case we
  * have \f$ A = V D V^{-1} \f$. This is called the eigendecomposition.
  *
  * The eigenvalues and eigenvectors of a matrix may be complex, even when the
  * matrix is real. However, we can choose real matrices \f$ V \f$ and \f$ D
  * \f$ satisfying \f$ A V = V D \f$, just like the eigendecomposition, if the
  * matrix \f$ D \f$ is not required to be diagonal, but if it is allowed to
  * have blocks of the form
  * \f[ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \f]
  * (where \f$ u \f$ and \f$ v \f$ are real numbers) on the diagonal.  These
  * blocks correspond to complex eigenvalue pairs \f$ u \pm iv \f$. We call
  * this variant of the eigendecomposition the pseudo-eigendecomposition.
  *
  * Call the function compute() to compute the eigenvalues and eigenvectors of
  * a given matrix. Alternatively, you can use the 
  * EigenSolver(const MatrixType&, bool) constructor which computes the
  * eigenvalues and eigenvectors at construction time. Once the eigenvalue and
  * eigenvectors are computed, they can be retrieved with the eigenvalues() and
  * eigenvectors() functions. The pseudoEigenvalueMatrix() and
  * pseudoEigenvectors() methods allow the construction of the
  * pseudo-eigendecomposition.
  *
  * The documentation for EigenSolver(const MatrixType&, bool) contains an
  * example of the typical use of this class.
  *
  * \note The implementation is adapted from
  * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain).
  * Their code is based on EISPACK.
  *
  * \sa MatrixBase::eigenvalues(), class ComplexEigenSolver, class SelfAdjointEigenSolver
  */
template <typename _MatrixType> class EigenSolver
{
public:
    /** \brief Synonym for the template parameter \p _MatrixType. */
    typedef _MatrixType MatrixType;

    enum
    {
        RowsAtCompileTime = MatrixType::RowsAtCompileTime,
        ColsAtCompileTime = MatrixType::ColsAtCompileTime,
        Options = MatrixType::Options,
        MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
        MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };

    /** \brief Scalar type for matrices of type #MatrixType. */
    typedef typename MatrixType::Scalar Scalar;
    typedef typename NumTraits<Scalar>::Real RealScalar;
    typedef Eigen::Index Index;  ///< \deprecated since Eigen 3.3

    /** \brief Complex scalar type for #MatrixType. 
      *
      * This is \c std::complex<Scalar> if #Scalar is real (e.g.,
      * \c float or \c double) and just \c Scalar if #Scalar is
      * complex.
      */
    typedef std::complex<RealScalar> ComplexScalar;

    /** \brief Type for vector of eigenvalues as returned by eigenvalues(). 
      *
      * This is a column vector with entries of type #ComplexScalar.
      * The length of the vector is the size of #MatrixType.
      */
    typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;

    /** \brief Type for matrix of eigenvectors as returned by eigenvectors(). 
      *
      * This is a square matrix with entries of type #ComplexScalar. 
      * The size is the same as the size of #MatrixType.
      */
    typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> EigenvectorsType;

    /** \brief Default constructor.
      *
      * The default constructor is useful in cases in which the user intends to
      * perform decompositions via EigenSolver::compute(const MatrixType&, bool).
      *
      * \sa compute() for an example.
      */
    EigenSolver() : m_eivec(), m_eivalues(), m_isInitialized(false), m_eigenvectorsOk(false), m_realSchur(), m_matT(), m_tmp() {}

    /** \brief Default constructor with memory preallocation
      *
      * Like the default constructor but with preallocation of the internal data
      * according to the specified problem \a size.
      * \sa EigenSolver()
      */
    explicit EigenSolver(Index size)
        : m_eivec(size, size), m_eivalues(size), m_isInitialized(false), m_eigenvectorsOk(false), m_realSchur(size), m_matT(size, size), m_tmp(size)
    {
    }

    /** \brief Constructor; computes eigendecomposition of given matrix. 
      * 
      * \param[in]  matrix  Square matrix whose eigendecomposition is to be computed.
      * \param[in]  computeEigenvectors  If true, both the eigenvectors and the
      *    eigenvalues are computed; if false, only the eigenvalues are
      *    computed. 
      *
      * This constructor calls compute() to compute the eigenvalues
      * and eigenvectors.
      *
      * Example: \include EigenSolver_EigenSolver_MatrixType.cpp
      * Output: \verbinclude EigenSolver_EigenSolver_MatrixType.out
      *
      * \sa compute()
      */
    template <typename InputType>
    explicit EigenSolver(const EigenBase<InputType>& matrix, bool computeEigenvectors = true)
        : m_eivec(matrix.rows(), matrix.cols()), m_eivalues(matrix.cols()), m_isInitialized(false), m_eigenvectorsOk(false), m_realSchur(matrix.cols()),
          m_matT(matrix.rows(), matrix.cols()), m_tmp(matrix.cols())
    {
        compute(matrix.derived(), computeEigenvectors);
    }

    /** \brief Returns the eigenvectors of given matrix. 
      *
      * \returns  %Matrix whose columns are the (possibly complex) eigenvectors.
      *
      * \pre Either the constructor 
      * EigenSolver(const MatrixType&,bool) or the member function
      * compute(const MatrixType&, bool) has been called before, and
      * \p computeEigenvectors was set to true (the default).
      *
      * Column \f$ k \f$ of the returned matrix is an eigenvector corresponding
      * to eigenvalue number \f$ k \f$ as returned by eigenvalues().  The
      * eigenvectors are normalized to have (Euclidean) norm equal to one. The
      * matrix returned by this function is the matrix \f$ V \f$ in the
      * eigendecomposition \f$ A = V D V^{-1} \f$, if it exists.
      *
      * Example: \include EigenSolver_eigenvectors.cpp
      * Output: \verbinclude EigenSolver_eigenvectors.out
      *
      * \sa eigenvalues(), pseudoEigenvectors()
      */
    EigenvectorsType eigenvectors() const;

    /** \brief Returns the pseudo-eigenvectors of given matrix. 
      *
      * \returns  Const reference to matrix whose columns are the pseudo-eigenvectors.
      *
      * \pre Either the constructor 
      * EigenSolver(const MatrixType&,bool) or the member function
      * compute(const MatrixType&, bool) has been called before, and
      * \p computeEigenvectors was set to true (the default).
      *
      * The real matrix \f$ V \f$ returned by this function and the
      * block-diagonal matrix \f$ D \f$ returned by pseudoEigenvalueMatrix()
      * satisfy \f$ AV = VD \f$.
      *
      * Example: \include EigenSolver_pseudoEigenvectors.cpp
      * Output: \verbinclude EigenSolver_pseudoEigenvectors.out
      *
      * \sa pseudoEigenvalueMatrix(), eigenvectors()
      */
    const MatrixType& pseudoEigenvectors() const
    {
        eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
        eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
        return m_eivec;
    }

    /** \brief Returns the block-diagonal matrix in the pseudo-eigendecomposition.
      *
      * \returns  A block-diagonal matrix.
      *
      * \pre Either the constructor 
      * EigenSolver(const MatrixType&,bool) or the member function
      * compute(const MatrixType&, bool) has been called before.
      *
      * The matrix \f$ D \f$ returned by this function is real and
      * block-diagonal. The blocks on the diagonal are either 1-by-1 or 2-by-2
      * blocks of the form
      * \f$ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \f$.
      * These blocks are not sorted in any particular order.
      * The matrix \f$ D \f$ and the matrix \f$ V \f$ returned by
      * pseudoEigenvectors() satisfy \f$ AV = VD \f$.
      *
      * \sa pseudoEigenvectors() for an example, eigenvalues()
      */
    MatrixType pseudoEigenvalueMatrix() const;

    /** \brief Returns the eigenvalues of given matrix. 
      *
      * \returns A const reference to the column vector containing the eigenvalues.
      *
      * \pre Either the constructor 
      * EigenSolver(const MatrixType&,bool) or the member function
      * compute(const MatrixType&, bool) has been called before.
      *
      * The eigenvalues are repeated according to their algebraic multiplicity,
      * so there are as many eigenvalues as rows in the matrix. The eigenvalues 
      * are not sorted in any particular order.
      *
      * Example: \include EigenSolver_eigenvalues.cpp
      * Output: \verbinclude EigenSolver_eigenvalues.out
      *
      * \sa eigenvectors(), pseudoEigenvalueMatrix(),
      *     MatrixBase::eigenvalues()
      */
    const EigenvalueType& eigenvalues() const
    {
        eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
        return m_eivalues;
    }

    /** \brief Computes eigendecomposition of given matrix. 
      * 
      * \param[in]  matrix  Square matrix whose eigendecomposition is to be computed.
      * \param[in]  computeEigenvectors  If true, both the eigenvectors and the
      *    eigenvalues are computed; if false, only the eigenvalues are
      *    computed. 
      * \returns    Reference to \c *this
      *
      * This function computes the eigenvalues of the real matrix \p matrix.
      * The eigenvalues() function can be used to retrieve them.  If 
      * \p computeEigenvectors is true, then the eigenvectors are also computed
      * and can be retrieved by calling eigenvectors().
      *
      * The matrix is first reduced to real Schur form using the RealSchur
      * class. The Schur decomposition is then used to compute the eigenvalues
      * and eigenvectors.
      *
      * The cost of the computation is dominated by the cost of the
      * Schur decomposition, which is very approximately \f$ 25n^3 \f$
      * (where \f$ n \f$ is the size of the matrix) if \p computeEigenvectors 
      * is true, and \f$ 10n^3 \f$ if \p computeEigenvectors is false.
      *
      * This method reuses of the allocated data in the EigenSolver object.
      *
      * Example: \include EigenSolver_compute.cpp
      * Output: \verbinclude EigenSolver_compute.out
      */
    template <typename InputType> EigenSolver& compute(const EigenBase<InputType>& matrix, bool computeEigenvectors = true);

    /** \returns NumericalIssue if the input contains INF or NaN values or overflow occurred. Returns Success otherwise. */
    ComputationInfo info() const
    {
        eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
        return m_info;
    }

    /** \brief Sets the maximum number of iterations allowed. */
    EigenSolver& setMaxIterations(Index maxIters)
    {
        m_realSchur.setMaxIterations(maxIters);
        return *this;
    }

    /** \brief Returns the maximum number of iterations. */
    Index getMaxIterations() { return m_realSchur.getMaxIterations(); }

private:
    void doComputeEigenvectors();

protected:
    static void check_template_parameters()
    {
        EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
        EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL);
    }

    MatrixType m_eivec;
    EigenvalueType m_eivalues;
    bool m_isInitialized;
    bool m_eigenvectorsOk;
    ComputationInfo m_info;
    RealSchur<MatrixType> m_realSchur;
    MatrixType m_matT;

    typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
    ColumnVectorType m_tmp;
};

template <typename MatrixType> MatrixType EigenSolver<MatrixType>::pseudoEigenvalueMatrix() const
{
    eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
    const RealScalar precision = RealScalar(2) * NumTraits<RealScalar>::epsilon();
    Index n = m_eivalues.rows();
    MatrixType matD = MatrixType::Zero(n, n);
    for (Index i = 0; i < n; ++i)
    {
        if (internal::isMuchSmallerThan(numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i)), precision))
            matD.coeffRef(i, i) = numext::real(m_eivalues.coeff(i));
        else
        {
            matD.template block<2, 2>(i, i) << numext::real(m_eivalues.coeff(i)), numext::imag(m_eivalues.coeff(i)), -numext::imag(m_eivalues.coeff(i)),
                numext::real(m_eivalues.coeff(i));
            ++i;
        }
    }
    return matD;
}

template <typename MatrixType> typename EigenSolver<MatrixType>::EigenvectorsType EigenSolver<MatrixType>::eigenvectors() const
{
    eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
    eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
    const RealScalar precision = RealScalar(2) * NumTraits<RealScalar>::epsilon();
    Index n = m_eivec.cols();
    EigenvectorsType matV(n, n);
    for (Index j = 0; j < n; ++j)
    {
        if (internal::isMuchSmallerThan(numext::imag(m_eivalues.coeff(j)), numext::real(m_eivalues.coeff(j)), precision) || j + 1 == n)
        {
            // we have a real eigen value
            matV.col(j) = m_eivec.col(j).template cast<ComplexScalar>();
            matV.col(j).normalize();
        }
        else
        {
            // we have a pair of complex eigen values
            for (Index i = 0; i < n; ++i)
            {
                matV.coeffRef(i, j) = ComplexScalar(m_eivec.coeff(i, j), m_eivec.coeff(i, j + 1));
                matV.coeffRef(i, j + 1) = ComplexScalar(m_eivec.coeff(i, j), -m_eivec.coeff(i, j + 1));
            }
            matV.col(j).normalize();
            matV.col(j + 1).normalize();
            ++j;
        }
    }
    return matV;
}

template <typename MatrixType>
template <typename InputType>
EigenSolver<MatrixType>& EigenSolver<MatrixType>::compute(const EigenBase<InputType>& matrix, bool computeEigenvectors)
{
    check_template_parameters();

    using numext::isfinite;
    using std::abs;
    using std::sqrt;
    eigen_assert(matrix.cols() == matrix.rows());

    // Reduce to real Schur form.
    m_realSchur.compute(matrix.derived(), computeEigenvectors);

    m_info = m_realSchur.info();

    if (m_info == Success)
    {
        m_matT = m_realSchur.matrixT();
        if (computeEigenvectors)
            m_eivec = m_realSchur.matrixU();

        // Compute eigenvalues from matT
        m_eivalues.resize(matrix.cols());
        Index i = 0;
        while (i < matrix.cols())
        {
            if (i == matrix.cols() - 1 || m_matT.coeff(i + 1, i) == Scalar(0))
            {
                m_eivalues.coeffRef(i) = m_matT.coeff(i, i);
                if (!(isfinite)(m_eivalues.coeffRef(i)))
                {
                    m_isInitialized = true;
                    m_eigenvectorsOk = false;
                    m_info = NumericalIssue;
                    return *this;
                }
                ++i;
            }
            else
            {
                Scalar p = Scalar(0.5) * (m_matT.coeff(i, i) - m_matT.coeff(i + 1, i + 1));
                Scalar z;
                // Compute z = sqrt(abs(p * p + m_matT.coeff(i+1, i) * m_matT.coeff(i, i+1)));
                // without overflow
                {
                    Scalar t0 = m_matT.coeff(i + 1, i);
                    Scalar t1 = m_matT.coeff(i, i + 1);
                    Scalar maxval = numext::maxi<Scalar>(abs(p), numext::maxi<Scalar>(abs(t0), abs(t1)));
                    t0 /= maxval;
                    t1 /= maxval;
                    Scalar p0 = p / maxval;
                    z = maxval * sqrt(abs(p0 * p0 + t0 * t1));
                }

                m_eivalues.coeffRef(i) = ComplexScalar(m_matT.coeff(i + 1, i + 1) + p, z);
                m_eivalues.coeffRef(i + 1) = ComplexScalar(m_matT.coeff(i + 1, i + 1) + p, -z);
                if (!((isfinite)(m_eivalues.coeffRef(i)) && (isfinite)(m_eivalues.coeffRef(i + 1))))
                {
                    m_isInitialized = true;
                    m_eigenvectorsOk = false;
                    m_info = NumericalIssue;
                    return *this;
                }
                i += 2;
            }
        }

        // Compute eigenvectors.
        if (computeEigenvectors)
            doComputeEigenvectors();
    }

    m_isInitialized = true;
    m_eigenvectorsOk = computeEigenvectors;

    return *this;
}

template <typename MatrixType> void EigenSolver<MatrixType>::doComputeEigenvectors()
{
    using std::abs;
    const Index size = m_eivec.cols();
    const Scalar eps = NumTraits<Scalar>::epsilon();

    // inefficient! this is already computed in RealSchur
    Scalar norm(0);
    for (Index j = 0; j < size; ++j) { norm += m_matT.row(j).segment((std::max)(j - 1, Index(0)), size - (std::max)(j - 1, Index(0))).cwiseAbs().sum(); }

    // Backsubstitute to find vectors of upper triangular form
    if (norm == Scalar(0))
    {
        return;
    }

    for (Index n = size - 1; n >= 0; n--)
    {
        Scalar p = m_eivalues.coeff(n).real();
        Scalar q = m_eivalues.coeff(n).imag();

        // Scalar vector
        if (q == Scalar(0))
        {
            Scalar lastr(0), lastw(0);
            Index l = n;

            m_matT.coeffRef(n, n) = Scalar(1);
            for (Index i = n - 1; i >= 0; i--)
            {
                Scalar w = m_matT.coeff(i, i) - p;
                Scalar r = m_matT.row(i).segment(l, n - l + 1).dot(m_matT.col(n).segment(l, n - l + 1));

                if (m_eivalues.coeff(i).imag() < Scalar(0))
                {
                    lastw = w;
                    lastr = r;
                }
                else
                {
                    l = i;
                    if (m_eivalues.coeff(i).imag() == Scalar(0))
                    {
                        if (w != Scalar(0))
                            m_matT.coeffRef(i, n) = -r / w;
                        else
                            m_matT.coeffRef(i, n) = -r / (eps * norm);
                    }
                    else  // Solve real equations
                    {
                        Scalar x = m_matT.coeff(i, i + 1);
                        Scalar y = m_matT.coeff(i + 1, i);
                        Scalar denom =
                            (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) + m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag();
                        Scalar t = (x * lastr - lastw * r) / denom;
                        m_matT.coeffRef(i, n) = t;
                        if (abs(x) > abs(lastw))
                            m_matT.coeffRef(i + 1, n) = (-r - w * t) / x;
                        else
                            m_matT.coeffRef(i + 1, n) = (-lastr - y * t) / lastw;
                    }

                    // Overflow control
                    Scalar t = abs(m_matT.coeff(i, n));
                    if ((eps * t) * t > Scalar(1))
                        m_matT.col(n).tail(size - i) /= t;
                }
            }
        }
        else if (q < Scalar(0) && n > 0)  // Complex vector
        {
            Scalar lastra(0), lastsa(0), lastw(0);
            Index l = n - 1;

            // Last vector component imaginary so matrix is triangular
            if (abs(m_matT.coeff(n, n - 1)) > abs(m_matT.coeff(n - 1, n)))
            {
                m_matT.coeffRef(n - 1, n - 1) = q / m_matT.coeff(n, n - 1);
                m_matT.coeffRef(n - 1, n) = -(m_matT.coeff(n, n) - p) / m_matT.coeff(n, n - 1);
            }
            else
            {
                ComplexScalar cc = ComplexScalar(Scalar(0), -m_matT.coeff(n - 1, n)) / ComplexScalar(m_matT.coeff(n - 1, n - 1) - p, q);
                m_matT.coeffRef(n - 1, n - 1) = numext::real(cc);
                m_matT.coeffRef(n - 1, n) = numext::imag(cc);
            }
            m_matT.coeffRef(n, n - 1) = Scalar(0);
            m_matT.coeffRef(n, n) = Scalar(1);
            for (Index i = n - 2; i >= 0; i--)
            {
                Scalar ra = m_matT.row(i).segment(l, n - l + 1).dot(m_matT.col(n - 1).segment(l, n - l + 1));
                Scalar sa = m_matT.row(i).segment(l, n - l + 1).dot(m_matT.col(n).segment(l, n - l + 1));
                Scalar w = m_matT.coeff(i, i) - p;

                if (m_eivalues.coeff(i).imag() < Scalar(0))
                {
                    lastw = w;
                    lastra = ra;
                    lastsa = sa;
                }
                else
                {
                    l = i;
                    if (m_eivalues.coeff(i).imag() == RealScalar(0))
                    {
                        ComplexScalar cc = ComplexScalar(-ra, -sa) / ComplexScalar(w, q);
                        m_matT.coeffRef(i, n - 1) = numext::real(cc);
                        m_matT.coeffRef(i, n) = numext::imag(cc);
                    }
                    else
                    {
                        // Solve complex equations
                        Scalar x = m_matT.coeff(i, i + 1);
                        Scalar y = m_matT.coeff(i + 1, i);
                        Scalar vr = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) +
                                    m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag() - q * q;
                        Scalar vi = (m_eivalues.coeff(i).real() - p) * Scalar(2) * q;
                        if ((vr == Scalar(0)) && (vi == Scalar(0)))
                            vr = eps * norm * (abs(w) + abs(q) + abs(x) + abs(y) + abs(lastw));

                        ComplexScalar cc = ComplexScalar(x * lastra - lastw * ra + q * sa, x * lastsa - lastw * sa - q * ra) / ComplexScalar(vr, vi);
                        m_matT.coeffRef(i, n - 1) = numext::real(cc);
                        m_matT.coeffRef(i, n) = numext::imag(cc);
                        if (abs(x) > (abs(lastw) + abs(q)))
                        {
                            m_matT.coeffRef(i + 1, n - 1) = (-ra - w * m_matT.coeff(i, n - 1) + q * m_matT.coeff(i, n)) / x;
                            m_matT.coeffRef(i + 1, n) = (-sa - w * m_matT.coeff(i, n) - q * m_matT.coeff(i, n - 1)) / x;
                        }
                        else
                        {
                            cc = ComplexScalar(-lastra - y * m_matT.coeff(i, n - 1), -lastsa - y * m_matT.coeff(i, n)) / ComplexScalar(lastw, q);
                            m_matT.coeffRef(i + 1, n - 1) = numext::real(cc);
                            m_matT.coeffRef(i + 1, n) = numext::imag(cc);
                        }
                    }

                    // Overflow control
                    Scalar t = numext::maxi<Scalar>(abs(m_matT.coeff(i, n - 1)), abs(m_matT.coeff(i, n)));
                    if ((eps * t) * t > Scalar(1))
                        m_matT.block(i, n - 1, size - i, 2) /= t;
                }
            }

            // We handled a pair of complex conjugate eigenvalues, so need to skip them both
            n--;
        }
        else
        {
            eigen_assert(0 && "Internal bug in EigenSolver (INF or NaN has not been detected)");  // this should not happen
        }
    }

    // Back transformation to get eigenvectors of original matrix
    for (Index j = size - 1; j >= 0; j--)
    {
        m_tmp.noalias() = m_eivec.leftCols(j + 1) * m_matT.col(j).segment(0, j + 1);
        m_eivec.col(j) = m_tmp;
    }
}

}  // end namespace Eigen

#endif  // EIGEN_EIGENSOLVER_H
